Can anybody help me with proving the following?
"Show that if a bipartite graph is $k$-connected, then it has a matching with $k$ edges."
Here's my attempt at using induction on $k$:
It's obviously true for 1-connected bipartite graphs.
Let $G$ be a $k$-connected bipartite graph. $G$ has an edge $uv$. Let H be a graph obtained from $G$ by removing vertuices $u,v$.
Here's my problem: I don't know how to prove that $H$ is $(k-1)$-connected.
If I know $H$ is $(k-1)$-connected then by induction hypothesis $H$ has a matching $M$ with $(k-1)$ edges. Then $M \cup \{ uv \}$ is a matching in $G$ with $k$ edges.
Or maybe you can prove it without induction?